THE IMPLEMENTATION AND ANALYSIS OF KALMAN FILTER PHASE
UNWRAPPING ALGORITHM OF INSAR
LIU Guo-lin*, Hao Hua-dong, Tao Qiu-xiang, Bai Ting-yi, Hu Le-yin
Geomatics College, Shandong University of Science and Technology, Qingdao 266510, Shandong, China
gliu@sdust.edu.cn
KEY WORDS: InSAR, Phase unwrapping, Kalman filter, Noise reduction, Algorithm implementation, Result analysis
ABSTRACT:
Phase unwrapping is the difficulty and hotspot in research because it is the key step in InSAR. The general phase unwrapping
methods need usually use some filtering algorithms to eliminate noise firstly, and then to phase unwrapping, so as to ensure the
quality of unwrapping. The Kalman filter overcomes this drawback, and transforms the phase unwrapping problem into state
estimate to deal with phase unwrapping and noise reduction at the same time.Choosing simulated data to processing, it is shown that
Kalman filter phase unwrapping method is very feasible and effective in the respect of unwrapping and noise restraining through the
analysis and evaluation of the results.
1. INTRODUCTION
problem, the real phase is solved through the establishment of
the dynamic equation and observation equation of phase. But
the biggest feature of this method is to achieve phase
unwrapping and simultaneous noise reduction, overcoming the
drawback that the general methods must be implemented prior
to the elimination of phase noise before phase unwrapping.
InSAR provides critical information about disaster and
emergency responses related to volcanic eruptions, earthquakes,
landslides, flooding, and high-resolution DEM generation, etc.
In order to infer the cause that is responsible for the observed
deformation or generate a DEM, InSAR images have to be
unwrapped. The general phase unwrapping algorithms are
broadly divided into two categories. The first is based on the
path following algorithms[1-3],such as Goldstein’s branch cut
algorithm. The second is based on the minimum norm
algorithms[4-7],such as minimum Lp-norm algorithm.The
former is a local algorithm, its advantages are that it can isolate
the phase branch points, prevent the local phase error in the
spread of the whole integral region, and gain the accurate
unwrapping phase in the region of better coherence. But it is
difficult to get the best integral path, and easily lead to errors
transmission or the isolated region under the strong noise
conditions.While the latter is a global algorithm, the excellence
is that operation is stable, and it do not need to identify the
residues. But they are "through" the residues instead of
"bypass" them, so it is likely to result in errors transmission.
The basic model and algorithm of Kalman filter are briefly
introducted in this paper, then it is to evaluate the performance
of the algorithm in two aspects of unwrapping and filtering
using simulated data for experiment.
2. KALMAN FILTER MODEL OF PHASE
UNWRAPPING
2.1 Observation Model
Consisting of in-phase and quadrature components of the
complex interferogram as two noisy observations of the true
interferometric phase[13]. In order to writing conveniently,
substituting again the n, m dependence by a 1-D k dependence,
we let
In addition to the above two categories algorithms, people have
taken more and more attention in using optimal estimation
algorithms to phase unwrapping, such as Network flow model
[8-10], Bayesian model [11], Kalman filter model [12-16] and
Grid-based filter model [17]. Although phase unwrapping
methods have developed rapidly in the past 20 years and
achieved fruitful results, but also be widely used. However, due
to the complexity of phase unwrapping and its affected by many
factors, phase unwrapping has still been the difficult and
research hotspot in InSAR. A large number of residues are
generated in phase unwrapping process because the
interferograms are impacted by noise. Thus it seriously affects
the quality of phase unwrapping, and even causes its failure.
Therefore, the general phase unwrapping methods must be
carried out to eliminate noise before unwrapping. Kalman filter
transforms the phase unwrapping issue into a state estimation
⎡ ⎧ z ( k ) ⎫⎤
⎬⎥
⎢Re ⎨
a( k ) ⎭⎥ ⎡cos(ϕ ( k ))⎤ ⎡ v1 (k ) ⎤
y (k ) = ⎢ ⎩
Δ
+
⎢ ⎧ z ( k ) ⎫⎥ ⎢⎣ sin(ϕ (k )) ⎥⎦ ⎢⎣v 2 (k )⎥⎦
⎬⎥
⎢ Im⎨
⎣ ⎩ a ( k ) ⎭⎦
= h( x (k )) + v(k )
(1)
ϕ (k ) is the true unambiguous phase, since there is no difference
between true and wrapped phase in the complex Domain.Thus,
any filter directly operating on the complex data, rather than on
the phases calculated from the complex values, will implicitly
not suffer from the wraparound effect.
2.2 State-Space Model
A simple but very effective state-space model for the wanted
unambiguous phase is given in [16]. Starting with the
* Geomatics college,Shandong University of Science and Technology, No.579 Qianwangang Road, Economic&Technical
Development Zone,Qingdao, Shandong Province, P.R.China;266510
30
interferometric phase in a discrete-time representation, we
assume that
∧
x1 (k + 1) = ϕ (k + 1) = x1 (k ) + u (k )
u (k ) = uˆ (k ) + w(k )
(2)
3. KALMAN FILTER AND PHASE UNWRAPPING
Established for the above mentioned state-space model, if the
statistical properties of noise is known, we can accord to the
following extended Kalman filter for state estimation [18]:
(a)
xˆ k ,k −1 = Φ k ,k −1 xˆ k −1,k −1 + q
(3)
Pk ,k −1 = Φ k ,k −1Pk −1ΦTk ,k −1 + Q
J k = Pk ,k −1 A ( Ak Pk ,k −1 A + R)
T
k
T
k
(4)
−1
(5)
xˆ k = xˆ k ,k = xˆ k ,k −1 + J k ( Lk − Ak xˆ k ,k −1 − r )
(6)
Pk = Pk ,k = ( I − J k Ak ) Pk ,k −1
(7)
where J k =gain matrix
xˆ k ,k −1 , Pk ,k −1 =predictive value and its covariance matrix,
respectively
xˆ k , k , Pk , k =filter value and its covariance matrix,
respectively
(b)
Ak = linearized observation matrix
Figure 1. (a) True phase surface(without noise)
(b) Kalman filter unwrapping phase surface
Ak is given by
As shown in Figure 1 (a), we simulate a phase surface. In
experiment, we deal with it to wrap phase (Figure 2 (a)),
d
h(x ) | xˆk ,k −1
Ak =
dx(k )
(8)
= [− sin( xˆ k ,k −1 ), cos( xˆ k ,k −1 )]
T
Phase slope estimation q ,the measurement noise covariance R ,
and the driving noise covariance Q can be calculated in [14].
The specific implementation of two-dimensional Kalman filter
phase unwrapping can be seen in [12] and [13].
4.
EXPERIMENTAL RESULTS AND ANALYSIS
The simulated data is used in two different experiments under
the different conditions of coherence. From the view of
visualization and quantitation, Kalman filter phase unwrapping
algorithm is analyzed and evaluated in two aspects of the filter
effect and unwrapping quality.
Figure 2. (a) Wrapped phase image(without noise). (b) Noisy
wrapped phase image(coherence=0.4). (c) Rewrapped phase
image of unwrapping result of Kalman filter. (d) Wrapped
phase image after ML filter.
4.1 Experiment I
adding phase noise which the coherent is 0.4, the noisy wrapped
phase can be seen in Figure 2 (b).
31
Filter method
Mean abs.phase
error(rad)
1.3169
1.0840
ML filter
Kalman filter
Mean square error
(rad2)
0.0115
0.0080
Table 1. Difference comparison between the results of two
filter methods and noisy wrapped phase (coherence=0.4)
Figure 3. (a) Unwrapping result of Goldstein’s branch cut
algorithm after ML filter. (b) Unwrapping result of Kalman
filter.
Visual evaluation of unwrapping quality: Contrasting Figure
1 (a) with (b), the Kalman filter does well in restoring a real
surface phase. Comparison of Figure 3 (a) and (b), it can be
seen unwrapping result of Goldstein’s branch cut algorithm
(Figure 3 (a)) is poorer than that of Kalman filter method
(Figure 3 (b)).Despite of prefiltering, the containing noise
causes branch cut method can not determine the correct branchcut, forming some isolated regions (dark blue spot regions in
Figure 3 (a)).
Using Goldstein’s branch cut method after prefiltering and
Kalman filter phase unwrapping algorithm to deal with
respectively, and the results are in Figure 3 (a) and (b). Here we
use Maximum Likelihood(ML) filter as a prefiltering approach.
The Kalman filter unwrapping phase surface is shown in Figure
1 (b). The rewrapped phase of unwrapping result of Kalman
filter is in Figure 2 (c); meanwhile, the result after doing a ML
filter to wrapped phase directly is in Figure 2 (d).
Phase
unwrapping
method
Goldstein’s
branch cut
algorithm
after ML
filter
Kalman filter
unwrapping
Algorithm
Max
abs.phase
error (rad)
Min
abs.phase
error (rad)
Mean
abs.phase
error (rad)
Mean
square
error
(rad2)
9.1480
0
0.6471
0.0025
3.3330
2.8312×
10-6
0.5017
0.0021
Table 2. Difference comparison between the results of two
unwrapping methods and true phase (without noise)
Quantitative evaluation of unwrapping quality: We subtract
true phase from unwrapping result of Kalman filter and branch
cut method separately, and then to compare their differences.
From Table 2, we can see although the min. abs. phase error of
Kalman filter method is very close to that of branch-cut method
after prefiltering, the max. abs. phase error, mean abs.phase
error and mean square error are less than prefiltering branch-cut
method. This shows that in the condition that the coherence is
0.4, even if it does prefiltering to noisy wrapped phase, the
result of branch-cut method is still unsatisfactory, but Kalman
filtering approach is relatively good.
Figure 4. The profile of the first 200 line corresponding to
Figure 2 (a) (b) (c) (d), respectively.
Visual evaluation of filter effect: By comparing Figure 2 (c)
with Figure 2 (b) and Figure 2 (d) separately, we can see the
fringes in Figure 2 (c) become more clear and bright. Figure 4 is
shown the profile of the first 200 line corresponding to Figure 2
(a) (b) (c) (d), respectively. The curves in Figure 4 (c) become
more smooth after contrasting Figure 4 (c) with Figure 4 (d).
This shows that compared with the ML filter, Kalman filter
deals with phase unwrapping and noise effective elimination
simultanously.
On the whole, the result of Kalman filter phase unwrapping is
satisfactory in poorer coherence.
4.2 Experiment II
Quantitative evaluation of filter effect: We subtract noisy
wrapped phase(coherence=0.4) from the filter result of Kalman
filter and ML filter separately, and then to compare their
differences. Note: The filter result of Kalman filter is the
rewrapped phase of unwrapping result of Kalman filter. From
Table 1, we can see that the mean abs.phase error and mean
square error of Kalman filter are both less than the ML filter. It
is indicated that when the coherence is 0.4, the Kalman filter is
a little better than the ML filter.
Adding phase noise which the coherence is 0.75 to the true
phase surface(Figure 1(a)), we get the noisy wrapped phase
(Figure 5 (b)). Using Goldstein’s branch cut method without
prefiltering, after prefiltering and Kalman filter phase
unwrapping algorithm to deal with respectively, the results are
32
Figure 6. (a)Wrapped phase image(without noise). (b)Noisy
wrapped phase image(coherence=0.75). (c)Rewrapped phase
image of unwrapping result of Kalman filter. (d)Wrapped
phase image after ML filter.
Figure 5. Some results. (a)True phase image(without noise).
(b)Noisy wrapped phase image(coherence=0.75). (c)Wrapped
phase image after ML filter. (d)Unwrapping result of
Goldstein’s branch cut algorithm without prefiltering.
(e)Unwrapping result of Goldstein’s branch cut algorithm after
ML filter. (f) Unwrapping phase image of Kalman filter.
in Figure 5 (d) , (e) and (f). Here we use ML filter as a
prefiltering approach.
Visual evaluation of unwrapping quality: Contrasting Figure
5 (d) with (e), it can be seen the unwrapping result of
Goldstein’s branch cut algorithm after prefiltering (Figure 5
(e))is better than that without prefiltering (Figure 5 (d)). This is
because without prefiltering, forming many isolated regions
(dark blue regions in Figure 5 (a)).
Quantitative evaluation of unwrapping quality: We subtract
true phase from the unwrapping result of branch cut method
without prefiltering, that of branch cut method after prefiltering,
that of Kalman filter separately, and then to compare their
differences. From Table 3, we can see branch cut method after
prefiltering is better than that without prefiltering in max. abs.
phase error, min. abs. phase error, mean abs.phase error and
mean square error. It is to say that in the conditions that the
coherence is 0.75, it is in favor to phase unwrapping by using
prefiltering to noisy wrapped phase, the result is better.
Although Kalman filtering approach is slightly
Phase unwrapping
method
Goldstein’s branch cut
algorithm without prefiltering
Goldstein’s branch cut
algorithm after ML filter
Kalman filter unwrapping
algorithm
Figure 7. The profiles of the first 200 line corresponding to
Figure 6 (a) (b) (c) (d), respectively.
larger than branch cut method after prefiltering in min. abs.
phase error and mean abs.phase error, it is smaller than that
in max. abs. phase error and mean square error.
Max abs.phase
error (rad)
Min abs.phase
error (rad)
Mean abs.phase
error (rad)
Mean square
error (rad2)
12.8180
0
0.7757
3.0301×10-3
3.6132
0
0.2479
9.6835×10-4
2.5645
1.1563×10-5
0.4198
9.1433×10-5
Table 3. Difference comparison between the results of three unwrapping methods and true phase (without noise)
Visual evaluation of filter effect: By comparing Figure 6 (d)
with 6 (b) and 6 (c) separately, we can see the fringes in Figure
6 (d) become more clear and bright. Figure 7 is the profile of
In general, the result of Kalman filter phase unwrapping and
branch cut method after prefiltering are close in better
coherence.
33
the first 200 line corresponding to Figure 6 (a) (b) (c) (d),
respectively. The curves in Figure 7 (d) become more smooth
after contrasting Fig 6 (c) with Figure 6 (d). This shows that
Kalman filter is poorer than ML filter in the condition that the
coherence is 0.75.
[9] M.Contantini, A.Farina, F.Zirilli,1999. A Fast Phase
Unwrapping Alorgrithm for SAR Interferometry. IEEE Trans.
Geosci. Remote Sens., 37(1),pp.452-460.
[10] Curtis W Chen, 2001. Statistical-cost network-flow
approaches to two-dimensional phase unwrapping for radar
interferometry . Stanford University, Stanford, California, USA.
5. CONCLUSION
Using simulated data to experiment, the results are showed that
we can choose Goldstein’s branch cut method after prefiltering
in the situation of better coherence; meanwhile, in the less
coherence case, we use Kalman filter algorithm. Because it does
unwrapping and filter at the same time, eliminating the noise
effectively, indicating its advantages in unwrapping and filter.
However, when the terrain is steep and slope is larger, the
unwrapping results will be bad and cause errors transmission.
Therefore, how the Kalman filter algorithm taking into account
the impact of the terrain will be a future research direction.
[11] Dias, J.M.B. Leitao, J.M.N,2001. InSAR phase
unwrapping: a Bayesian approach. In: Geoscience and Remote
Sensing Symposium, IGARSS'01, Sydney, Australia,Vol.1,
pp.396-400.
[12] R.Krämer, O.Loffeld,1997.Presentation of an improved
Phase Unwrapping Algorithm based on Kalman filters
combined with local slope estimation. In: Proceedings of
Fringe'96 ESA Workshop Applications of ERS SAR
Interferometry,Zurich, Switzerland,pp.253-260.
[13] O.Loffeld, R.Krämer,1999.Phase unwrapping for SAR
interferometry—A data fusion approach by Kalman filtering.In:
Geoscience and Remote Sensing Symposium.IGARSS
'99,Vol.3,pp.1715-1717.
ACKNOWLEDGEMENTS
The research is supported by the National Natural Science
Foundation of China(40874001), Key Laboratory of Geoinformatics of State Bureau of Surveying and Mapping(200810),
Special Project Fund of Taishan Scholars of Shandong
Province(TSXZ-0520). The authors would like to express a
special thank to Batuhan Osmanoglu in University of Miami for
his help and advice.
[14] O.Loffeld, H.Nies, S.Knedlik, and Y.Wang, 2008. Phase
unwrapping for SAR Interferometry—A data fusion approach
by kalman filtering. In: IEEE Transactions on Geoscience and
Remote Sensing, Vol.46, pp. 47-58.
[15] B. Osmanoglu, T. Dixon, S. Wdowinski, J. Biggs,
2009.InSAR phase unwrapping based on Extended Kalman
Filtering. In: Radarcon09, Pasadena, CA.
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